Let $h(x)=8\log_5(x)$. Find $h'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{8}{x\log_5(x)}$ (Choice B) B $\dfrac{8\ln(5)}{\ln(x)}$ (Choice C) C $\dfrac{8}{\ln(5)x}$ (Choice D) D $\dfrac{8}{x}$
Solution: The expression for $h(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative of the function as shown below. $\begin{aligned} h'(x)&=\dfrac{d}{dx}[8\log_5(x)] \\\\ &=8\dfrac{d}{dx}[\log_5(x)] \\\\ &=8\cdot\dfrac{1}{\ln(5)x} \\\\ &=\dfrac{8}{\ln(5)x} \end{aligned}$ In conclusion, $h'(x)=\dfrac{8}{\ln(5)x}$.